Question Video: Identifying the Relationship between Two Lines Mathematics • 8th Grade

Fill in the blank: In the diagram, we can conclude that 𝐿₁ is ＿ 𝐿₂.

04:26

Video Transcript

Fill in the blank. In the diagram, we can conclude that 𝐿 sub one is what 𝐿 sub two.

In the diagram, we can observe that there are four different lines which intersect at various points. The two lines in particular that we need to consider are the lines 𝐿 sub one and 𝐿 sub two, which are marked here in pink. The missing word or words in the statement will be something that describes the relationship between these two lines. It might be useful to consider what possible relationships two lines in two dimensions could have.

So let’s begin with the word “perpendicular.” Perpendicular lines cross at right angles. Or the lines could be parallel. That would mean they would always be the same distance apart and never meet. Next, we could also describe two lines as simply intersecting. So the lines will meet or cross, but not necessarily at right angles. Finally, another possible option is that the lines are coincident. That means they would share all points and would in fact lie on top of one another.

By looking at the diagram, we could rule out two of these possible options, because 𝐿 sub one and 𝐿 sub two don’t cross at 90 degrees and they appear as two distinct lines. Notice that if two lines are not parallel, then they would be intersecting at some point, even if we don’t see it on the diagram. The most likely situation is that these two lines are parallel. But we can’t just give this as the answer without proving it in some way.

Let’s recall the theorem that if corresponding angles, alternate interior angles, or alternate exterior angles formed by a transversal cutting two lines are congruent, then the lines cut by the transversal are parallel. So let’s see what we can tell from the given angle measures in the figure.

We are given three angle measures. But none of these are equal. So we’ll have to see if there are any other angle measures we can determine. One possible option would be to see if we can find the measure of this angle marked in blue, since it would potentially be an alternate interior angle to the angle whose measure is given as 100 degrees. We can define this unknown angle measure as 𝑥 degrees.

Now, consider that the angle measure of 𝑥 degrees lies within this triangle with another angle of measure 30 degrees and a third unknown angle, which we can call 𝑦 degrees. The angle measures of 𝑦 degrees and 130 degrees lie on a straight line. So these angle measures must sum to 180 degrees, which means that 𝑦 degrees is equal to 50 degrees.

We can now use the property that the sum of the interior angle measures in a triangle is 180 degrees to help us find the value of 𝑥. The three angle measures of 30 degrees, 50 degrees, and 𝑥 degrees must sum to 180 degrees. And by simplifying the left-hand side and subtracting 80 degrees from both sides, we have that 𝑥 degrees is 100 degrees.

Now, remember, we wanted to find the value of this angle measure because we wanted to check if there are congruent angles. And we have found that a pair of alternate angles formed by a transversal cutting two lines are congruent. Therefore, the two lines are parallel. So we can complete the blank with the words “parallel to,” since we have proved that in the diagram, line 𝐿 sub one is parallel to line 𝐿 sub two.